On the Langlands parameters for \(A_q(\lambda)\) for classical simple Lie groups (Q2713106)

Let \(G\) be a classical noncompact simple (real) Lie group with finite center, \(\mathfrak g_0\) its Lie algebra with complexification \(\mathfrak g\), and \(K\) a maximal compact subgroup of \(G\). Suppose that \(\text{rank}(G) = \text{rank}(K)\) and that \(G/K\) is not Hermitian symmetric. A Cartan involution \(\theta\) of \(\mathfrak g_0\) being given, let \(\mathfrak q\) be a \(\theta\)-stable parabolic subalgebra of \(\mathfrak g\), \(L \subset K\) the normalizer of \(\mathfrak q\) in \(G\), and \(\lambda\) the unique weight of a one-dimensional representation of \(L\). We denote by \(V\) the irreducible subquotient of the Vogan-Zuckerman \((\mathfrak g,K)\)-module \(A_{\mathfrak q}(\lambda)\) [\textit{A. W. Knapp} and \textit{D. A. Vogan}, Cohomological induction and unitary representations, Princeton Univ. Press, Princeton, NJ (1995; Zbl 0863.22011)] which contains the unique minimal \(K\)-type of \(A_{\mathfrak q}(\lambda)\) [\textit{D. A. Vogan}, Ann. Math. (2) 109, 1-60 (1979; Zbl 0424.22010)]. A conjectural method aimed at finding the Langlands parameters of \(V\) was suggested by Knapp and shown to work if \(\mathfrak g_0 \cong \mathfrak s\mathfrak o(2m,2n) (m > 1, n > 1)\) [\textit{A. W. Knapp}, Represent. Theory 1, 1-24 (1997; Zbl 0887.22019)]. It is proved in this paper that Knapp's method can also be applied to every classical noncompact simple Lie group with finite center.